Not the same as the bike stations you're talking about, but Washington DC just because the first city in the U.S. with SmartBike bike rentals.

https://www.smartbikedc.com/default.asp

You pay $40 per year for membership and they send you a card which you take to one of their locations. You swipe the card, take a bike and ride it for up to 3 hours, then return it to any one of their locations. The bikes are designed for people to ride them in regular clothes without getting dirty (bit fenders and chain guards).